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Properties

Label 1-35e2-1225.13-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.997 - 0.0679i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.722 + 0.691i)2-s + (−0.919 + 0.393i)3-s + (0.0448 − 0.998i)4-s + (0.393 − 0.919i)6-s + (0.657 + 0.753i)8-s + (0.691 − 0.722i)9-s + (−0.691 − 0.722i)11-s + (0.351 + 0.936i)12-s + (0.990 − 0.134i)13-s + (−0.995 − 0.0896i)16-s + (0.998 − 0.0448i)17-s + i·18-s + (−0.809 − 0.587i)19-s + (0.998 + 0.0448i)22-s + (−0.351 + 0.936i)23-s + (−0.900 − 0.433i)24-s + ⋯
L(s)  = 1  + (−0.722 + 0.691i)2-s + (−0.919 + 0.393i)3-s + (0.0448 − 0.998i)4-s + (0.393 − 0.919i)6-s + (0.657 + 0.753i)8-s + (0.691 − 0.722i)9-s + (−0.691 − 0.722i)11-s + (0.351 + 0.936i)12-s + (0.990 − 0.134i)13-s + (−0.995 − 0.0896i)16-s + (0.998 − 0.0448i)17-s + i·18-s + (−0.809 − 0.587i)19-s + (0.998 + 0.0448i)22-s + (−0.351 + 0.936i)23-s + (−0.900 − 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.997 - 0.0679i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.997 - 0.0679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6663720150 + 0.02265236827i\)
\(L(\frac12)\) \(\approx\) \(0.6663720150 + 0.02265236827i\)
\(L(1)\) \(\approx\) \(0.5715512861 + 0.1325804656i\)
\(L(1)\) \(\approx\) \(0.5715512861 + 0.1325804656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.722 + 0.691i)T \)
3 \( 1 + (-0.919 + 0.393i)T \)
11 \( 1 + (-0.691 - 0.722i)T \)
13 \( 1 + (0.990 - 0.134i)T \)
17 \( 1 + (0.998 - 0.0448i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.351 + 0.936i)T \)
29 \( 1 + (0.963 + 0.266i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.351 - 0.936i)T \)
41 \( 1 + (-0.858 - 0.512i)T \)
43 \( 1 + (0.974 - 0.222i)T \)
47 \( 1 + (0.880 + 0.473i)T \)
53 \( 1 + (-0.998 - 0.0448i)T \)
59 \( 1 + (-0.995 - 0.0896i)T \)
61 \( 1 + (0.550 - 0.834i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.990 - 0.134i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.880 - 0.473i)T \)
89 \( 1 + (0.134 - 0.990i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.92374211421870186270350643944, −20.58215935342118274619994608555, −19.309306596558273661975371009959, −18.735579683798007799146840681362, −18.208378346212701261114673759393, −17.44495521295328269023771827206, −16.73557586318939930524496086594, −16.10342744726615712172263252646, −15.23266060832301241090691718168, −13.82068521879464007304752721759, −13.1085795267384113861551770449, −12.209624835719881742326232830955, −11.955741445606263079172790676985, −10.66429154566377648150574735536, −10.47292697408654387330075598621, −9.5609162836642904849782594927, −8.247434559293569197413496115608, −7.88428534167531224565169463751, −6.74186619680088815373864837953, −6.0696920759594928781027907051, −4.81526273641671490664896262459, −4.04166207546222521890728655394, −2.76888587869485632653049544754, −1.796199982584299402585026885007, −0.89810742901053314097623388431, 0.535927360682577666220990031909, 1.52001075277969248634240122574, 3.09808495910829877737025106258, 4.29390281489316073921950831274, 5.31670807105150012395486354021, 5.85626794587446807119943373628, 6.61029955195703467371689028562, 7.57698261085745314947766735933, 8.46458991754587064702647209035, 9.22738551923186715021614049544, 10.28766902541896697642921910580, 10.68191515550450415440498182817, 11.45294674205424587018091370475, 12.45455189568530243095018797005, 13.53385971864898655324731845232, 14.28168755837343402334242787060, 15.46973692666512372510121930199, 15.839900382618028247900402715275, 16.42029270372932584459213885325, 17.45873740211748505555172532394, 17.720947322731012138483078950618, 18.80304034839778534590121778435, 19.13964796628788400153942943214, 20.420930259530179109752564367279, 21.16786035775116621913354271821

Graph of the $Z$-function along the critical line